Traveling wave solution and qualitative behavior of fractional stochastic Kraenkel–Manna–Merle equation in ferromagnetic materials

The main purpose of this article is to investigate the qualitative behavior and traveling wave solutions of the fractional stochastic Kraenkel–Manna–Merle equations, which is commonly used to simulate the zero conductivity nonlinear propagation behavior of short waves in saturated ferromagnetic materials. Firstly, fractional stochastic Kraenkel–Manna–Merle equations are transformed into ordinary differential equations by using the traveling wave transformation. Secondly, the phase portraits, sensitivity analysis, and Poincaré sections of the two-dimensional dynamic system and its perturbation system of ordinary differential equations are drawn. Finally, the traveling wave solutions of fractional stochastic Kraenkel–Manna–Merle equations are obtained based on the analysis theory of planar dynamical system. Moreover, the obtained three-dimensional graphs of random solutions, two-dimensional graphs of random solutions, and three-dimensional graphs of deterministic solutions are drawn.

In this study, the fractional stochastic Kraenkel-Manna-Merle equations are presented as follows 30 where the magnetization and the external magnetic field related to the ferrite are represented by ϕ = ϕ(t, x) and ψ = ψ(t, x) , respectively.D α x stands for the conformable fractional derivative.κ is the coefficient of the damping.σ represents the noise intensity.B stands for the Brownian motion, and B t = ∂B ∂t .In 30 , Wael W. Mohammed et al. constructed the traveling wave solution of Eq. (1.1) by the Mapping method.The main purpose of this article is to study the phase diagram and traveling wave solutions of Eq. (1.1) by the theory of dynamical systems.At the same time, by adding small perturbations to the two-dimensional dynamical system, its chaotic behavior, sensitivity, and Poincaré sections are considered. (1.1)

OPEN
School of Electronic Information and Electrical Engineering, Chengdu University, Chengdu 610106, People's Republic of China.email: 18180693508@163.com The remaining sections in this article are arranged as follows: In Sect.2, the phase portraits of the dynamical system and its perturbed system are discussed.In Sect.3, the solutions of Eq. (1.1) are constructed by the method of analyzing planar dynamical systems.Finally, a brief conclusion is given.

Mathematical derivation
When κ = 0 , the wave transformation is considered here �(ξ ) and �(ξ ) are real functions.k 1 and k 2 stand for nonzero constants.

Chaotic behaviors
Therefore, system (2.6) can be transformed into the following two-dimensional disturbance system with perturbation term where f (ξ ) = A cos(kξ) and f (ξ ) = Ae −0.05ξ are the perturbed term.A represent the amplitude of system (2.9).k stands for the frequency of system (2.9).

Remark 2.5
Here, we have used the Maple 2022 mathematical software to draw two-dimensional diagrams, three-dimensional diagrams, sensitivity analysis, and Paincaré section of the disturbance system (2.9) (see Figs. 2  and 3).The disturbance systems we are considering are periodic disturbances and small disturbance systems.At the same time, we also considered the 2D phase diagram, 3D phase diagram, and sensitivity analysis of the disturbance system (2.9) under different initial values.

Numerical simulations
In this section, we plotted the solutions ψ 1 (t, x) and ψ 2 (t, x) including three-dimensional random graphs, two- dimensional random graphs, and three-dimensional deterministic graphs by setting different parameters and Maple 2022 mathematical software as shown in Figs. 4 and 5. ψ 1 (t, x) is a Jacobian function solution.ψ 2 (t, x) is a bell shaped solitary wave solution.

Conclusion
This article uses the theory of dynamical systems to study the traveling wave solutions and qualitative behavior of Eq. (1.1).Compared with reference 30 , this article not only obtained the traveling wave solution of Eq. (1.1), but also analysed the chaotic behavior, sensitivity analysis, and Poincaré sections by adding small perturbations.
In this article, we consider two different forms of disturbance systems.On the one hand, we consider periodic perturbation system.On the other hand, we also considered the dynamic behavior of small periodic disturbances.In order to facilitate readers' understanding of the solution of Eq. (1.1), we have drawn three-dimensional and www.nature.com/scientificreports/two-dimensional graphs of the solutions ψ 1 (t, x) and ψ 2 (t, x) , as well as three-dimensional graphs without considering the influence of random factors.In future research, our focus will be on the study of traveling wave solutions and dynamic behavior of more complex fractional order stochastic partial differential equations.